On the stability of the Ginzburg–Landau vortex

We introduce a functional framework taylored to investigate the minimality and stability properties of the Ginzburg-Landau vortex of degree one on the whole plane. We prove that a renormalized Ginzburg-Landau energy is well-defined in that framework and that the vortex is its unique global minimizer up to the invariances by translation and phase shift. Our main result is a nonlinear coercivity estimate for the renormalized energy around the vortex, from which we can deduce its orbital stability as a solution to the Gross-Pitaevskii equation, the natural Hamiltonian evolution equation associated to the Ginzburg-Landau energy.

[1]  Israel Michael Sigal,et al.  Ginzburg-Landau equation I. Static vortices , 1997 .

[2]  Jean-Claude Saut,et al.  Travelling Waves for the Gross-Pitaevskii Equation II , 2007, 0711.2408.

[3]  Philippe Gravejat,et al.  Asymptotic stability of the black soliton for the Gross–Pitaevskii equation , 2015 .

[4]  Daniel Spirn,et al.  Refined Jacobian Estimates and Gross–Pitaevsky Vortex Dynamics , 2008 .

[5]  Petru Mironescu,et al.  Les minimiseurs locaux pour l'équation de Ginzburg-Landau sont à symétrie radiale , 1996 .

[6]  F. Béthuel,et al.  A remark on the Cauchy problem for the 2D Gross-Pitaevskii equation with nonzero degree at infinity , 2007, Differential and Integral Equations.

[7]  Robert L. Jerrard,et al.  Vortex dynamics for the Ginzburg-Landau-Schrodinger equation , 1997 .

[8]  J. Xin,et al.  On the Incompressible Fluid Limit and the Vortex Motion Law of the Nonlinear Schrödinger Equation , 1999 .

[9]  M. Weinstein.  Lyapunov stability of ground states of nonlinear dispersive evolution equations , 1986 .

[10]  Petru Mironescu,et al.  On the Stability of Radial Solutions of the Ginzburg-Landau Equation , 1995 .

[11]  Charles M. Elliott,et al.  Shooting method for vortex solutions of a complex-valued Ginzburg–Landau equation , 1994, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[12]  Daniel Spirn,et al.  Refined Jacobian estimates for Ginzburg-Landau functionals , 2007 .

[13]  J. Saut,et al.  Existence and properties of travelling waves for the Gross-Pitaevskii equation , 2009, 0902.3804.

[14]  M. Pino,et al.  Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy's type inequality , 2004 .

[15]  Michael I. Weinstein,et al.  Modulational Stability of Ground States of Nonlinear Schrödinger Equations , 1985 .

[16]  Philippe Gravejat,et al.  Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation , 2012, 1206.2221.

[17]  Robert L. Jerrard,et al.  On the NLS dynamics for infinite energy vortex configurations on the plane , 2008 .